# Fungrim entry: 2488bb

$J^{(r)}_{\nu}\!\left(z\right) = \frac{1}{{2}^{r}} \sum_{k=0}^{r} {\left(-1\right)}^{k} {r \choose k} J_{\nu + 2 k - r}\!\left(z\right)$
Assumptions:$\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}$
Alternative assumptions:$\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}$
TeX:
J^{(r)}_{\nu}\!\left(z\right) = \frac{1}{{2}^{r}} \sum_{k=0}^{r} {\left(-1\right)}^{k} {r \choose k} J_{\nu + 2 k - r}\!\left(z\right)

\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
BesselJDerivative$J^{(r)}_{\nu}\!\left(z\right)$ Differentiated Bessel function of the first kind
Pow${a}^{b}$ Power
Sum$\sum_{n} f\!\left(n\right)$ Sum
Binomial${n \choose k}$ Binomial coefficient
BesselJ$J_{\nu}\!\left(z\right)$ Bessel function of the first kind
ZZ$\mathbb{Z}$ Integers
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("2488bb"),
Formula(Equal(BesselJDerivative(nu, z, r), Mul(Div(1, Pow(2, r)), Sum(Mul(Mul(Pow(-1, k), Binomial(r, k)), BesselJ(Sub(Add(nu, Mul(2, k)), r), z)), Tuple(k, 0, r))))),
Variables(nu, z, r),
Assumptions(And(Element(nu, ZZ), Element(z, CC), Element(r, ZZGreaterEqual(0))), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-17 11:32:46.829430 UTC